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2015, 35(10): 5107-5131. doi: 10.3934/dcds.2015.35.5107

## Monotonicity, asymptotics and uniqueness of travelling wave solution of a non-local delayed lattice dynamical system

 1 Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, China 2 College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China

Received  May 2014 Revised  January 2015 Published  April 2015

A delayed lattice dynamical system with non-local diffusion and interaction is considered in this paper. The exact asymptotics of the wave profile at both wave tails is derived, and all the wave profiles are shown to be strictly increasing. Moreover, we prove that the wave profile with a given admissible speed is unique up to translation. These results generalize earlier monotonicity, asymptotics and uniqueness results in the literature.
Citation: Zhaoquan Xu, Jiying Ma. Monotonicity, asymptotics and uniqueness of travelling wave solution of a non-local delayed lattice dynamical system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5107-5131. doi: 10.3934/dcds.2015.35.5107
##### References:
 [1] M. Aguerrea, C. Gomez and S. Trofimchuk, On uniqueness of semi-wavefronts: Diekmann-Kaper theory of a nonlinear convolution equation re-visited,, Math. Ann., 354 (2012), 73. doi: 10.1007/s00208-011-0722-8. [2] P. W. Bates and A. Chmaj, A discrete convolution model for phase transition,, Arch. Ration. Mech. Anal., 150 (1999), 281. doi: 10.1007/s002050050189. [3] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method,, Bol. Soc. Brasil. Math., 22 (1991), 1. doi: 10.1007/BF01244896. [4] X. Chen and J.-S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics,, Math. Ann., 326 (2003), 123. doi: 10.1007/s00208-003-0414-0. [5] X. Chen, S.-C. Fu and J.-S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices,, SIAM J. Math. Anal., 38 (2006), 233. doi: 10.1137/050627824. [6] X. Chen and J.-S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations,, J. Differential Equations, 184 (2002), 549. doi: 10.1006/jdeq.2001.4153. [7] J. Coville, On uniqueness and monotonicity of non-local reaction diffusion equation,, Ann. Mat. Pura Appl., 185 (2006), 461. doi: 10.1007/s10231-005-0163-7. [8] S.-N. Chow, J. Mallet-Paret and W. X. Shen, Travelling waves in lattice dynamical systems,, J. Differential Equations, 149 (1998), 248. doi: 10.1006/jdeq.1998.3478. [9] J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations,, Proc. Amer. Math. Soc., 132 (2004), 2433. doi: 10.1090/S0002-9939-04-07432-5. [10] O. Diekmann and H. Kaper, On the bounded solutions of a nonlinear convolution equation,, Nonlinear Anal. TMA., 2 (1978), 721. doi: 10.1016/0362-546X(78)90015-9. [11] J. Fang, J. Wei and X.-Q. Zhao, Uniqueness of traveling waves for nonlocal lattice equations,, Proc. Amer. Math. Soc., 139 (2010), 1361. doi: 10.1090/S0002-9939-2010-10540-3. [12] J. Fang, J. Wei and X.-Q. Zhao, Spreading speed and travelling waves for non-monotone time-delayed lattice equations,, Proc. Roy. Math. Soc. Lond. A., 466 (2010), 1919. doi: 10.1098/rspa.2009.0577. [13] J.-S. Guo and Y.-C. Lin, Traveling wave solution for a lattice dynamical system with convolution type nonlinearity,, Discrete Contin. Dyn. Syst., 32 (2012), 101. doi: 10.3934/dcds.2012.32.101. [14] X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with application,, Comm. Pure Appl. Math., 60 (2007), 1. doi: 10.1002/cpa.20154. [15] S. W. Ma and X. F. Zou, Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay,, J. Differential Equations, 217 (2005), 54. doi: 10.1016/j.jde.2005.05.004. [16] S. W. Ma, P. X. Weng and X. F. Zou, Asymptotic speed of propagation and traveling wavefronts in a non-local delayed lattice differential equation,, Nonlinear Analysis, 65 (2006), 1858. doi: 10.1016/j.na.2005.10.042. [17] H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spred and traveling waves for integral equations and delayed reaction-diffusion models,, J. Differential Equations, 195 (2003), 430. doi: 10.1016/S0022-0396(03)00175-X. [18] D. V. Widder, The Laplace Transform,, Princeton University Press, (1941). [19] P. X. Weng, H. X. Huang and J. H. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction,, IMA J. Appl. Math., 68 (2003), 409. doi: 10.1093/imamat/68.4.409. [20] J. H. Wu and X. F. Zou, Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations,, J. Differential Equations, 135 (1997), 315. doi: 10.1006/jdeq.1996.3232. [21] Z. X. Yu, Uniqueness of critical traveling waves for nonlocal lattice equations with delays,, Proc. Amer. Math. Soc., 140 (2012), 3853. doi: 10.1090/S0002-9939-2012-11225-0. [22] P. A. Zhang and W. T. Li, Moonotonicity and uniqueness of traveling waves for a reaction-diffusion model with a quiescent stage,, Nonlinear Anal. TMA., 72 (2010), 2178. doi: 10.1016/j.na.2009.10.016. [23] B. Zinner, G. Harris and W. Hudson, Traveling wavefronts for the discrete Fisher's equation,, J. Differential Equations, 105 (1993), 46. doi: 10.1006/jdeq.1993.1082.

show all references

##### References:
 [1] M. Aguerrea, C. Gomez and S. Trofimchuk, On uniqueness of semi-wavefronts: Diekmann-Kaper theory of a nonlinear convolution equation re-visited,, Math. Ann., 354 (2012), 73. doi: 10.1007/s00208-011-0722-8. [2] P. W. Bates and A. Chmaj, A discrete convolution model for phase transition,, Arch. Ration. Mech. Anal., 150 (1999), 281. doi: 10.1007/s002050050189. [3] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method,, Bol. Soc. Brasil. Math., 22 (1991), 1. doi: 10.1007/BF01244896. [4] X. Chen and J.-S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics,, Math. Ann., 326 (2003), 123. doi: 10.1007/s00208-003-0414-0. [5] X. Chen, S.-C. Fu and J.-S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices,, SIAM J. Math. Anal., 38 (2006), 233. doi: 10.1137/050627824. [6] X. Chen and J.-S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations,, J. Differential Equations, 184 (2002), 549. doi: 10.1006/jdeq.2001.4153. [7] J. Coville, On uniqueness and monotonicity of non-local reaction diffusion equation,, Ann. Mat. Pura Appl., 185 (2006), 461. doi: 10.1007/s10231-005-0163-7. [8] S.-N. Chow, J. Mallet-Paret and W. X. Shen, Travelling waves in lattice dynamical systems,, J. Differential Equations, 149 (1998), 248. doi: 10.1006/jdeq.1998.3478. [9] J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations,, Proc. Amer. Math. Soc., 132 (2004), 2433. doi: 10.1090/S0002-9939-04-07432-5. [10] O. Diekmann and H. Kaper, On the bounded solutions of a nonlinear convolution equation,, Nonlinear Anal. TMA., 2 (1978), 721. doi: 10.1016/0362-546X(78)90015-9. [11] J. Fang, J. Wei and X.-Q. Zhao, Uniqueness of traveling waves for nonlocal lattice equations,, Proc. Amer. Math. Soc., 139 (2010), 1361. doi: 10.1090/S0002-9939-2010-10540-3. [12] J. Fang, J. Wei and X.-Q. Zhao, Spreading speed and travelling waves for non-monotone time-delayed lattice equations,, Proc. Roy. Math. Soc. Lond. A., 466 (2010), 1919. doi: 10.1098/rspa.2009.0577. [13] J.-S. Guo and Y.-C. Lin, Traveling wave solution for a lattice dynamical system with convolution type nonlinearity,, Discrete Contin. Dyn. Syst., 32 (2012), 101. doi: 10.3934/dcds.2012.32.101. [14] X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with application,, Comm. Pure Appl. Math., 60 (2007), 1. doi: 10.1002/cpa.20154. [15] S. W. Ma and X. F. Zou, Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay,, J. Differential Equations, 217 (2005), 54. doi: 10.1016/j.jde.2005.05.004. [16] S. W. Ma, P. X. Weng and X. F. Zou, Asymptotic speed of propagation and traveling wavefronts in a non-local delayed lattice differential equation,, Nonlinear Analysis, 65 (2006), 1858. doi: 10.1016/j.na.2005.10.042. [17] H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spred and traveling waves for integral equations and delayed reaction-diffusion models,, J. Differential Equations, 195 (2003), 430. doi: 10.1016/S0022-0396(03)00175-X. [18] D. V. Widder, The Laplace Transform,, Princeton University Press, (1941). [19] P. X. Weng, H. X. Huang and J. H. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction,, IMA J. Appl. Math., 68 (2003), 409. doi: 10.1093/imamat/68.4.409. [20] J. H. Wu and X. F. Zou, Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations,, J. Differential Equations, 135 (1997), 315. doi: 10.1006/jdeq.1996.3232. [21] Z. X. Yu, Uniqueness of critical traveling waves for nonlocal lattice equations with delays,, Proc. Amer. Math. Soc., 140 (2012), 3853. doi: 10.1090/S0002-9939-2012-11225-0. [22] P. A. Zhang and W. T. Li, Moonotonicity and uniqueness of traveling waves for a reaction-diffusion model with a quiescent stage,, Nonlinear Anal. TMA., 72 (2010), 2178. doi: 10.1016/j.na.2009.10.016. [23] B. Zinner, G. Harris and W. Hudson, Traveling wavefronts for the discrete Fisher's equation,, J. Differential Equations, 105 (1993), 46. doi: 10.1006/jdeq.1993.1082.
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